Sampling & Sampling Distributions Chapter 7 MSIS 111 Prof. Nick Dedeke

Learning Objectives Determine when to use sampling instead of a census. Distinguish between random and nonrandom sampling. Decide when and how to use various sampling techniques. Understand the impact of the Central Limit Theorem on statistical analysis. Use the sampling distributions of and. x  p

What is Sampling? Sampling is the process that is used to select entities that are representative of a given population. A sample is a set of entities that has been drawn from a given population using sampling methods.

Reasons for Sampling Sampling can save money. Sampling can save time. For given resources, sampling can broaden the scope of the data set. Whenever a testing process involves destruction of objects, sampling is mandatory. If assessment of total population is impossible; sampling is the only option.

Reasons for Taking a Census Eliminate the possibility that, by chance, a random sample taken may not be representative of the population. For the safety of the consumer.

Population Frame Population Frame: A list, map, directory, or other source used to represent the population. The population frame is used to select samples not the target population. Overregistration -- the frame contains all members of the target population and some additional elements Underregistration -- the frame does not contain all members of the target population. Q. Is the complete US national phone book a good frame for US census?

Random Versus Nonrandom Sampling Random sampling Every unit of the population has the same probability of being included in the sample. A chance mechanism is used in the selection process. Eliminates bias in the selection process Also known as probability sampling Nonrandom Sampling Every unit of the population does not have the same probability of being included in the sample. Not appropriate data collection methods for most statistical methods Also known as nonprobability sampling

Random Sampling Techniques Simple Random Sample Stratified Random Sample Proportionate (% of the sample taken from each stratum is proportionate to the % that each stratum is within the whole population) Disproportionate (when the % of the sample taken from each stratum is not proportionate to the % that each stratum is within the whole population) Sampling error occurs when, by chance, the sample selected is not representative of the population.

Simple Random Sample Number each frame unit from 1 to N. Use a random number table or a random number generator to select n distinct numbers between 1 and N, inclusively. Easier to perform for small populations Cumbersome for large populations

Simple Random Sample: Numbered Population Frame 01 Alaska Airlines 02 Alcoa 03 Ashland 04 Bank of America 05 BellSouth 06 Chevron 07 Citigroup 08 Clorox 09 Delta Air Lines 10 Disney 11 DuPont 12 Exxon Mobil 13 General Dynamics 14 General Electric 15 General Mills 16 Halliburton 17 IBM 18 Kellog 19 KMart 20 Lowe’s 21 Lucent 22 Mattel 23 Mead 24 Microsoft 25 Occidental Petroleum 26 JCPenney 27 Procter & Gamble 28 Ryder 29 Sears 30 Time Warner

Simple Random Sampling: Random Number Table N = 30 (count two digits in random no. table); n =6 01 Alaska Airlines 02 Alcoa 03 Ashland 04 Bank of America 05 BellSouth 06 Chevron 07 Citigroup 08 Clorox 09 Delta Air Lines 10 Disney 11 DuPont 12 Exxon Mobil 13 General Dynamics 14 General Electric 15 General Mills 16 Halliburton 17 IBM 18 Kellog 19 KMart 20 Lowe’s 21 Lucent 22 Mattel 23 Mead 24 Microsoft 25 Occidental Petroleum 26 JCPenney 27 Procter & Gamble 28 Ryder 29 Sears 30 Time Warner

Simple Random Sample: Sample Members 01 Alaska Airlines 02 Alcoa 03 Ashland 04 Bank of America 05 BellSouth 06 Chevron 07 Citigroup 08 Clorox 09 Delta Air Lines 10 Disney 11 DuPont 12 Exxon Mobil 13 General Dynamics 14 General Electric 15 General Mills 16 Halliburton 17 IBM 18 Kellog 19 KMart 20 Lowe’s 21 Lucent 22 Mattel 23 Mead 24 Microsoft 25 Occidental Petroleum 26 JCPenney 27 Procter & Gamble 28 Ryder 29 Sears 30 Time Warner N = 30 n = 6

Stratified Random Sample Population is divided into nonoverlapping subpopulations called strata. A random sample is selected from each stratum. Potential for reducing sampling error Stratification examples By geographic region By age By income By political party affiliation

Stratified Random Sample: Population of FM Radio Listeners years old (homogeneous within) (alike) years old (homogeneous within) (alike) years old (homogeneous within) (alike) Heterogeneous (different) between Heterogeneous (different) between Stratified by Age

Stratified Sampling Excel Example

Systematic Sampling Convenient and relatively easy to administer Population elements are an ordered sequence (at least, conceptually). The first sample element is selected randomly from the first k population elements. Thereafter, sample elements are selected at a constant interval, k, from the ordered sequence frame. k = N n, where: n= sample size N= population size k= size of selection interval

Cluster Sampling Population is divided into nonoverlapping clusters or areas. Each cluster is a miniature, or microcosm, of the population. A subset of the clusters is selected randomly for the sample. If the number of elements in the subset of clusters is larger than the desired value of n, these clusters may be subdivided to form a new set of clusters and subjected to a random selection process. Each cluster is heterogeneous within! Used when one needs to test markets and areas rather than just respondents.

Cluster Sampling u Advantages More convenient for geographically dispersed populations Reduced travel costs to contact sample elements Simplified administration of the survey Unavailability of sampling frame prohibits using other random sampling methods u Disadvantages Statistically less efficient when the cluster elements are similar Costs and problems of statistical analysis are greater than for simple random sampling.

Cluster Sampling San Jose Boise Phoenix Denver Cedar Rapids Buffalo Louisville Atlanta Portland Milwaukee Kansas City San Diego Tucson Grand Forks Fargo Sherman- Dension Odessa- Midland Cincinnati Pittsfield

Nonrandom Sampling Convenience Sampling: sample elements are selected for the convenience of the researcher Judgment Sampling: sample elements are selected by the judgment of the researcher Quota Sampling: sample elements are selected until the quota controls are satisfied Snowball Sampling: survey subjects are selected based on referral from other survey respondents

Errors u Data from nonrandom samples are not appropriate for analysis by inferential statistical methods. u Sampling Error occurs when the sample is not representative of the population. u Nonsampling Errors all errors that are not sampling errors. Such as: Missing Data, Recording, Data Entry, and Analysis Errors Poorly conceived concepts, unclear definitions, and defective questionnaires Response errors occur when people so not know, will not say, or overstate in their answers

Sampling Distribution of Proper analysis and interpretation of a sample statistic requires knowledge of its distribution. Process of Inferential Statistics x

Sampling Distribution of Notice something about sampling. The mean will always change even if we change one objects in a random sample. Or say it another way, every population has multitudes of ways in which randon samples can be selected and with that a large number of possible sample means. So, it is of interest to see how the possible sample means are distributed. x

Distribution of a Small Finite Population Population Histogram Frequency N = 8 54, 55, 59, 63, 68, 69, 70

Sample Space for n = 2 with Replacement SampleMeanSampleMeanSampleMeanSampleMean 1(54,54)54.017(59,54)56.533(64,54)59.049(69,54)61.5 2(54,55)54.518(59,55)57.034(64,55)59.550(69,55)62.0 3(54,59)56.519(59,59)59.035(64,59)61.551(69,59)64.0 4(54,63)58.520(59,63)61.036(64,63)63.552(69,63)66.0 5(54,64)59.021(59,64)61.537(64,64)64.053(69,64)66.5 6(54,68)61.022(59,68)63.538(64,68)66.054(69,68)68.5 7(54,69)61.523(59,69)64.039(64,69)66.555(69,69)69.0 8(54,70)62.024(59,70)64.540(64,70)67.056(69,70)69.5 9(55,54)54.525(63,54)58.541(68,54)61.057(70,54) (55,55)55.026(63,55)59.042(68,55)61.558(70,55) (55,59)57.027(63,59)61.043(68,59)63.559(70,59) (55,63)59.028(63,63)63.044(68,63)65.560(70,63) (55,64)59.529(63,64)63.545(68,64)66.061(70,64) (55,68)61.530(63,68)65.546(68,68)68.062(70,68) (55,69)62.031(63,69)66.047(68,69)68.563(70,69) (55,70)62.532(63,70)66.548(68,70)69.064(70,70)70.0 Shows what happens if we take all the possible samples of size n = 2 from the population and calculate the sample mean.

Distribution of the Sample Means Sampling Distribution Histogram Frequency

1,800 Randomly Selected Values from an Exponential Distribution X FrequencyFrequency

Means of 60 Samples (n = 2) from an Exponential Distribution FrequencyFrequency x

Means of 60 Samples (n = 5) from an Exponential Distribution FrequencyFrequency x

Means of 60 Samples (n = 30) from an Exponential Distribution FrequencyFrequency x

1,800 Randomly Selected Values from a Uniform Distribution X FrequencyFrequency

Means of 60 Samples (n = 2) from a Uniform Distribution FrequencyFrequency x

Means of 60 Samples (n = 5) from a Uniform Distribution FrequencyFrequency x

Means of 60 Samples (n = 30) from a Uniform Distribution FrequencyFrequency x

For sufficiently large sample sizes (n  30), The distribution of sample means, is approximately normal; The mean of this distribution is equal to , the population mean; and Its standard deviation is, Regardless of the shape of the population distribution. Central Limit Theorem n  x

Exponential Population n = 2n = 5n = 30 Distribution of Sample Means for Various Sample Sizes Uniform Population n = 2n = 5n = 30

Sampling from a Normal Population The distribution of sample means is normal for any sample size.

Examples in Excel